1. Stuff you MUST know Cold for the AP Calculus Exam
in the morning of Wednesday, May 7, 2014.
Written by Sean Bird
Updated May 4, 2013 by Ms. Acciardo

2. Curve sketching and analysis
y = f(x) must be continuous at each:
critical point: = 0 or undefined. And don’t forget endpoints
local minimum: goes (–,0,+) or (–,und,+) or > 0
local maximum: goes (+,0,–) or (+,und,–) or < 0
point of inflection: concavity changes
goes from (+,0,–), (–,0,+), (+,und,–), or (–,und,+)

3. Basic Derivatives

4. Basic Derivatives

5. More Derivatives
Recall “change of base”

6. Differentiation Rules
Chain Rule
Product Rule
Quotient Rule

7. The Fundamental Theorem of Calculus
Corollary to FTC

8. Intermediate Value Theorem
If the function f(x) is continuous on [a, b], and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a, b) such that f(c) = y.
Mean Value Theorem . .
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b), then there is at least one number x = c in (a, b) such that

9. Mean Value Theorem & Rolle’s Theorem
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b),
then there is at least one number x = c
in (a, b) such that
If the function f(x) is continuous on [a, b], AND the first derivative exists on the interval (a, b),
AND f(a) = f(b),
then there is at least one number x = c
in (a, b) such that f '(c) = 0.

10. Approximation Methods for Integration
Trapezoidal Rule
Simpson’s Rule
Simpson only works for
Even sub intervals (odd data points) 1/3 ( )

11. Theorem of the Mean Value i.e. AVERAGE VALUE
If the function f(x) is continuous on [a, b] and the first derivative exists on the interval (a, b), then there exists a number x = c on (a, b) such that
This value f(c) is the “average value” of the function on the interval [a, b].

12. Solids of Revolution and friends
Disk Method
Arc Length *bc topic
Washer Method
General volume equation (not rotated)

13. Distance, Velocity, and Acceleration
(position)
average velocity =
acceleration =
(velocity)
speed =
*velocity vector =
displacement =
*bc topic

14. Values of Trigonometric Functions for Common Angles
π/3 = 60° π/6 = 30° θ
sin θ
cos θ
tan θ 0° 1
sine
,30°
cosine
37°
3/5
4/5
3/4
,45° 1
53°
4/5
3/5
4/3
Pi/3 is 60 degrees
Pi/6 is 30 degrees
,60°
,90° 1 ∞
π,180° –1

15. Trig Identities
Double Argument

16. Trig Identities
Double Argument
Pythagorean
sine
cosine

17. Slope – Parametric & Polar
Parametric equation
Given a x(t) and a y(t) the slope is
Polar
Slope of r(θ) at a given θ is
What is y equal to in terms of r and θ ? x?

18. Polar Curve For a polar curve r(θ), the AREA inside a “leaf” is
(Because instead of infinitesimally small rectangles, use triangles)
where θ1 and θ2 are the “first” two times that r = 0.
We know arc length l = r θ
and

19. l’Hôpital’s Rule If
then

20. Integration by Parts L I P E T Logarithm Inverse Polynomial
We know the product rule L I P E T
Logarithm
Inverse
Polynomial
Exponential
Trig
Antiderivative product rule
(Use u = LIPET)
e.g.
Let u = ln x dv = dx
du = dx v = x

21. Maclaurin Series A Taylor Series about x = 0 is called Maclaurin.
If the function f is “smooth” at x = a, then it can be approximated by the nth degree polynomial